Enter names of editors and roles here. Be sure to read the home page for role choices. This is now a 25 point assignment.
The video should include all 6 members of this wiki page and one concept should be taught or demonstrated.
Vocabulary-Katie Conner
Real-Life Applications/Problem: Matthew Aguilar
Editor: Cheyenne Bohlen
Notes 7.1-7.3-Haley Brandt
Notes 7.4-7.6- Austin Gibbons
Example Problems- Cody Winsor 7.1 nth Roots and Rational exponents An nth root is the same concept as a square root with different numbers.
Ex: 3√8=2
The mathematic formula of an nth root is n√a=b. In this example 3 would be n, 8 would be a, and 2 would be b.
Real nth Roots n is a positive integer and a is a real number · If n is odd, then a has one real nth root. n√a=a1/n · If n is even and a >0, then a has two real nth roots. ±n√a=±a1/n · If n is even and a=0, then a has one nth root. n√0=01/n=0 · If n is even and a<0, then a has no real nth roots.
Ex:
Rational Exponents Let a1/n be an nth root of a and let m be a positive integer. · am/n=(a1/n)m=(n√a)m · a-m/n=1/(am/n)=1/(a1/n)m, a≠0
Ex:
7.2 Properties of Rational Exponents a and b are both real numbers and m and n are rational numbers.
Product of Powers Property
am•an=am+n
Power of a Power Property
(am)n=amn
Power of a Product Property
(ab)m=ambm
Negative Exponent Property
a-m=1/am
Quotient of Powers Property
am/an=am-n
Power of a Quotient Property
(a/b)m=am/bm
Real Life Application-
The equation f = 440 x 2^(n/12) can be used to find the frequencies of the musical range of an instrument, suchy as a trumpet or guitar.
"n" is based on A-440 which is the middle key on a piano, and the "A" key repeats every 12 notes.
This is what gives the "n/12" in the equation.
7.3 Power Functions and Function Operations Let f and g be two functions. The new function, h, can be made by combining f and g in any of four ways, addition, subtraction, division, and multiplication.
Addition
h(x)=f(x)+g(x)
Subtraction
h(x)=f(x)-g(x)
Multiplication
h(x)=f(x)∙g(x)
Division
h(x)=f(x)/g(x)
Composition of Two Functions The compostion of the function f with the function g is h(x)=f(g(x)). The domain of h is the set of all x-values such that x is in the domain of g and g(x)is in the domain of f.
Real Life Application-
By measuring the size of a fossilized footprint of a dinosaur, scientists can methematically determine the approximate height of the dinosaur that left the print based off of the the ratios that are known because of fossilized bones. These ratios are most often expotential.
7.4 Inverse Functions
Functions f and g are inverses of each other provided: f(g(x)) = x and g(f(x)) = x
The function g is denoted by f to the -1 power, read as " f inverse."
Horizontal Line Test
If no horizontal line intersects the graph of a function f more than once, then the inverse of f is itself a function.
Real Life Application-
One can use an equation such as h= 0.9 x (200-a) to determine a boweling handicap to level the ability range between differing players.
"h" is the handicap while "a" is a player's average score.
If their average happens to be over 200, their average would be 0.
Due to technical difficulties - the video will be shown during class.
Per Mrs. Keener
7.5 Graphing Square Root and Cube Root Functions
The graph of y = a times the square root of x, starts at the orgin and passes through the point (1,a). The graph of y = a times the cube root of x, passes through the orgin and the points (-1,-a) and (1,a).
Graphs of Radical Functions
To graph y = a times the square root of x-h, then plus k, or y = a times the cube root of x-h, then plus k, follow these steps.
STEP 1- Sketch the graph of y = a times the square root of x or y = a times the cube root of x.
STEP 2- Shift the graph h units horizontally and k units vertically.
Real Life Application-
If the horsepower within a drag race car is know, then one can determine the speed it will be expected to go in a certain distance.
s= 14.8 x p^(1/3) is an example model of this where "s" is the top speed while "p" is the horsepower of the car.
7.6 Solving Radical Equations
In order to solve a radical equation you must eliminate the radicals or rational exponents and be left with a polynomial equation. The key step is to raise each side of the equation to the same power.
Powers property of equality- If a = b, then a to the nth power = b to the nth power.
Isolate the radical expression on one side of the equation, then solve the new equation using standard procedures.
Real Life Application-
The Beaufort Wind Scale consists of an equation B= 1.69x (s + 4.45)^0.5 -3.49 to determine the approximate force and effect of the wind based on its speed.
"B" is the Beaufort Number, which ranges from 0 to 12 to determine the description of the weather. (0 = calm where smoke rises vertically, 12= hurricane with destruction)
"s" is the speed of the wind in miles per hour.
Vocabulary for Chapter Seven Section 7.1 Nth root of a-
If b^n=a, then b is the nth root of a. Index
- -n is the index number Section 7.2 Simplest Form
-to get a radical in this form you must apply the properties of radicals and remove any perfect nth powers (other than 1) and rationalize and denominators. Like Radicals
-expressions that have the same index and radicand. Section 7.3 Power Function
-common type of function which has the form y=ax^b where a is a real number and b is a rational number. Section 7.4 Inverse Relation
-maps the output values back to their original input values. Section 7.5 Radical Functions
-A function that contains a radical. Section 7.6 Radical Equation -an equation that contains radicals or rational exponents. Extraneous Solution
-a false solution.
Section 7.7 Statistics
-numerical values used to summarize and compare sets of data. Measures of Central Tendency
-three commonly used statistics: mean, median, and mode. Mean
- the average of n numbers is the sum of the numbers divided by n. Median
-the median of n numbers is the middle number when the numbers are written in order. Mode
-the mode of n numbers is the number or numbers that occur most frequently. Measures of Dispersion
-statistic that shows how spread out the data are. Range
-the difference between the greatest and least data values. Standard Deviation
-describes the typical difference between the mean and a data value. Box-and-Whisker Plot
-one type of statistical graph. A “box” encloses the middle half of the data set and the “whiskers” extend to the minimum and maximum data values.
Lower Quartile
-is the median of the lower half. Upper Quartile
-is the median of the upper half. Histogram
-Type of bar graph where data are grouped into intervals of equal width. Frequency
-the number of data values in each interval. Frequency Distribution
-shows the frequency of each interval.
Enter names of editors and roles here. Be sure to read the home page for role choices. This is now a 25 point assignment.
The video should include all 6 members of this wiki page and one concept should be taught or demonstrated.Vocabulary-Katie Conner
Real-Life Applications/Problem: Matthew Aguilar
Editor: Cheyenne Bohlen
Notes 7.1-7.3-Haley Brandt
Notes 7.4-7.6- Austin Gibbons
Example Problems- Cody Winsor
7.1 nth Roots and Rational exponents
An nth root is the same concept as a square root with different numbers.
Ex: 3√8=2
The mathematic formula of an nth root is n√a=b. In this example 3 would be n, 8 would be a, and 2 would be b.
Real nth Roots
n is a positive integer and a is a real number
· If n is odd, then a has one real nth root. n√a=a1/n
· If n is even and a >0, then a has two real nth roots. ±n√a=±a1/n
· If n is even and a=0, then a has one nth root. n√0=01/n=0
· If n is even and a<0, then a has no real nth roots.
Ex:
Rational Exponents
Let a1/n be an nth root of a and let m be a positive integer.
· am/n=(a1/n)m=(n√a)m
· a-m/n=1/(am/n)=1/(a1/n)m, a≠0
Ex:
7.2 Properties of Rational Exponents
a and b are both real numbers and m and n are rational numbers.
The equation f = 440 x 2^(n/12) can be used to find the frequencies of the musical range of an instrument, suchy as a trumpet or guitar.
"n" is based on A-440 which is the middle key on a piano, and the "A" key repeats every 12 notes.
This is what gives the "n/12" in the equation.
7.3 Power Functions and Function Operations
Let f and g be two functions. The new function, h, can be made by combining f and g in any of four ways, addition, subtraction, division, and multiplication.
Composition of Two Functions
The compostion of the function f with the function g is h(x)=f(g(x)). The domain of h is the set of all x-values such that x is in the domain of g and g(x)is in the domain of f.
Real Life Application-
By measuring the size of a fossilized footprint of a dinosaur, scientists can methematically determine the approximate height of the dinosaur that left the print based off of the the ratios that are known because of fossilized bones. These ratios are most often expotential.
7.4 Inverse Functions
Functions f and g are inverses of each other provided: f(g(x)) = x and g(f(x)) = x
The function g is denoted by f to the -1 power, read as " f inverse."
Horizontal Line Test
If no horizontal line intersects the graph of a function f more than once, then the inverse of f is itself a function.
Real Life Application-
One can use an equation such as h= 0.9 x (200-a) to determine a boweling handicap to level the ability range between differing players.
"h" is the handicap while "a" is a player's average score.
If their average happens to be over 200, their average would be 0.
Due to technical difficulties - the video will be shown during class.
Per Mrs. Keener
7.5 Graphing Square Root and Cube Root Functions
The graph of y = a times the square root of x, starts at the orgin and passes through the point (1,a).
The graph of y = a times the cube root of x, passes through the orgin and the points (-1,-a) and (1,a).
Graphs of Radical Functions
To graph y = a times the square root of x-h, then plus k, or y = a times the cube root of x-h, then plus k, follow these steps.
STEP 1- Sketch the graph of y = a times the square root of x or y = a times the cube root of x.
STEP 2- Shift the graph h units horizontally and k units vertically.
Real Life Application-
If the horsepower within a drag race car is know, then one can determine the speed it will be expected to go in a certain distance.
s= 14.8 x p^(1/3) is an example model of this where "s" is the top speed while "p" is the horsepower of the car.
7.6 Solving Radical Equations
In order to solve a radical equation you must eliminate the radicals or rational exponents and be left with a polynomial equation. The key step is to raise each side of the equation to the same power.
Powers property of equality- If a = b, then a to the nth power = b to the nth power.
Isolate the radical expression on one side of the equation, then solve the new equation using standard procedures.
Real Life Application-
The Beaufort Wind Scale consists of an equation B= 1.69x (s + 4.45)^0.5 -3.49 to determine the approximate force and effect of the wind based on its speed.
"B" is the Beaufort Number, which ranges from 0 to 12 to determine the description of the weather. (0 = calm where smoke rises vertically, 12= hurricane with destruction)
"s" is the speed of the wind in miles per hour.
Vocabulary for Chapter Seven
Section 7.1
Nth root of a-
If b^n=a, then b is the nth root of a.
Index
-
Section 7.2
Simplest Form
-to get a radical in this form you must apply the properties of radicals and remove any perfect nth powers (other than 1) and rationalize and denominators.
Like Radicals
-expressions that have the same index and radicand.
Section 7.3
Power Function
-common type of function which has the form y=ax^b where a is a real number and b is a rational number.
Section 7.4
Inverse Relation
-maps the output values back to their original input values.
Section 7.5
Radical Functions
-A function that contains a radical.
Section 7.6
Radical Equation
-an equation that contains radicals or rational exponents.
Extraneous Solution
-a false solution.
Section 7.7
Statistics
-numerical values used to summarize and compare sets of data.
Measures of Central Tendency
-three commonly used statistics: mean, median, and mode.
Mean
- the average of n numbers is the sum of the numbers divided by n.
Median
-the median of n numbers is the middle number when the numbers are written in order.
Mode
-the mode of n numbers is the number or numbers that occur most frequently.
Measures of Dispersion
-statistic that shows how spread out the data are.
Range
-the difference between the greatest and least data values.
Standard Deviation
-describes the typical difference between the mean and a data value.
Box-and-Whisker Plot
-one type of statistical graph. A “box” encloses the middle half of the data set and the “whiskers” extend to the minimum and maximum data values.
Lower Quartile
-is the median of the lower half.
Upper Quartile
-is the median of the upper half.
Histogram
-Type of bar graph where data are grouped into intervals of equal width.
Frequency
-the number of data values in each interval.
Frequency Distribution
-shows the frequency of each interval.